Units, Magnitude, and Notation

1. Units, Magnitude, and Notation We use SI (Système International) Units: You must know how to convert mi, in, ft, yd → mm, m, km, etc.!! Many stupid mistakes have been made by getting this wrong!

2. Waves In a lossless medium, one may write the current y as A = wave amplitude T = time period l = spatial wavelength f0 = reference phase We may define the total phase by writing This is the general form forany kind of wave, includingwater waves (Ulaby’s example) } There are 4 basic wave quantities in a lossless medium

3. Waves Ulaby When f0 = 0 • The phase changes by 2p in one period T or one wavelength l. • In Fig. 1-12, the wave moves in the +x direction with a velocity up = l/T.

4. Waves Phase velocity up: As time t increases, position x must increase to keep the phase constant. Since f is constant, we have So, a point of constant phase, the phase front, moves with velocity

5. Waves Other quantities (derived from the fundamental 4): f = frequency = 1/T (Hz = s–1) w= angular frequency = 2p f (rad/s = s–1) b = wavenumber = 2p /l (rad/m = m–1) In terms of these quantities, we have and

6. Waves Ulaby The phase offset: and (x = 0) Another useful representation: Letting R = Acosf0 , I = Asinf0 This representation will be useful when we discuss phasors

7. Waves With Loss: a = attenuation coefficient A A A Plot of y(x) = 10 exp (–0.2 x) cos(p x) amperes The envelope is bounded between 10 exp (–0.2 x)and its mirror image –10 exp (–0.2 x) A Ulaby (modified) With loss, there are 5 fundamental quantities

8. Waves With Loss: Ulaby Example 1-2 Question: A laser beam propagating through the atmosphere is characterized by an electric field intensity given by where x is the distance from the source in meters. Determine (a) the direction of wave travel, (b) the wave velocity, and (c) the wave amplitude at a distance of 200 m Solution: (a) Since the coefficients of t and x have the opposite sign, the wave propagates in the +x direction. (b) We find that which is (of course) the speed of light c in the vacuum or air. (c) At x = 200 m, the amplitude of E(x,t) is

9. Dispersion Relations • Dispersion relations: b(w) and a(w) are functions of f = w/2p • Calculating the dispersion relations is an important part of EM theory! • In a homogeneous, isotropic medium, this is straightforward • homogeneous = the same at all points in space • istropic = the same in all orientations (no strains; no crystal structure) • We calculate b(w) and a(w) from e(w) and m(w) • We will do this when we discuss plane waves • In an inhomogeneous, isotropic medium, we must account for geometry • e(w) → e(w,r) and m(w) → m(w,r) • The dispersion relations are determined by geometry as well as frequency • There can be multiple solutions at one frequency • We will do this for simple geometries As a consequence: The 5 fundamental quantities  3 independent quantities

10. Dispersion Relations • Dispersion relations: b(w) and a(w) are functions of f = w/2p • Calculating the dispersion relations is an important part of EM theory! • In an anisotropic medium, this becomes complex • D(r,w) = e(r,w)E(r,w) → D(r,w) = E(r,w)•E(r,w) ; • B(r,w) = m(r,w)H(r,w) → B(r,w) = M(r,w)•H(r,w) • where E(r,w) and M(r,w) are 33 matrices* • We will not discuss anisotropic media in this course • This discussion assumes that the medium is linear(Waves at different frequencies do not interact) • All media become linear when the wave amplitudes A are small enough *Strictly speaking, these are second-order tensors

11. Electromagnetic Spectrum Optical Fibers and Devices Optical fibers and devices typically operatein the range 0.8 mm to 1.8 mm with 1.5 mmcorresponding to minimum loss. Ulaby (modified) Note that the minimum loss band in atmospere is at visible light! Thus, it is sensible that this is the band of electromagnetic radiationthat our eyes see!

12. Electromagnetic Spectrum Ulaby The wavelength is tied to the frequency through the dispersionrelations.

13. Spectral Analysis Which frequencies are present? Following Paul, we consider a digital signal with a rise and fall time that is short compared to the base signal and generates high frequency components. The highest frequency is given approximately by 1/tmax. With a periodically repeatingsignal, the only frequencies thatappear are multiples of thebaseband f0 = 1/T , where T isthe period. Paul Fig. 1-3 a/b

14. Spectral Analysis Why are high harmonics / large frequency spreads bad? (1) The high harmonics lead to undesired electronic coupling in digital systems. (2) The high harmonics lead to pulse spreading in communications systems because different frequencies have different values of up. 1 1 0 1 0 Chromatic Dispersion: After spreading a factor of 5 in this return-to-zeroexample, the digital 1’scannot be distinguished fromthe digital 0’s 1 1 0 1 0

15. Complex Numbers and Phasors A complex numberz is written: We also write x = Re(z), y = Im(z) In polar form, we have: where |z| is the magnitude and q is the phase. From Euler’s identity, we find Graphical representationof the relationship betweenrectangular and polarcoordinates Ulaby

16. Complex Numbers and Phasors • The complex conjugatez* is defined: • Mathematical operations: • Equality: • Addition: • Multiplication: • Division: • Powers:

17. Complex Numbers and Phasors • The complex conjugatez* is defined: • Mathematical operations: • Logarithm: where n is any integer

18. –2 3 Re –3 I –4 V Complex Numbers and Phasors Working with phasors: Ulaby Example 1-3 Question: Given two complex numbers, V = 3 – j4 and I = –2 – j3, (a) Express V and I in polar form, and find (b) VI, (c) VI*, (d) V / I, (e) I½ Answer: (a) V is in the fourth quadrant and I is in the third quadrant. (See figure.) Im Angles are only unique to within 2p Based on Ulaby Fig. 1-18

19. Complex Numbers and Phasors • Why do we work with complex numbers? • The linear integro-differential equations that describe circuits and electromagnetic waves — and in fact any waves — become much easier to solve! • The concept of a phasor plays a key • Consider a simple RC circuit with a voltage source • Kirchoff’s voltage law implies • Direct time domain solution is a bit messy and involves “guessing” the integral. • The phasor approach is simpler and deductive (time domain equation) Ulaby Fig. 1-19

20. Complex Numbers and Phasors • Step 1: Adopt a cosine reference • Step 2: Express time-dependent variables as phasors • In general, we write any time-dependent variable z(t) as where is time-independent and is referred to as a phasor • In our particular case, we write where Ulaby Fig. 1-19

21. Complex Numbers and Phasors • Step 2: Express time-dependent variables as phasors (continued) • We now writeThe goal is to solve for , knowing , which will allow us to find i(t) • We will make use of two important properties • Step 3: Recast the equation in phasor form • This equation holds at all points in time if and only if (phasor/frequency/Fourier domain)

22. Complex Numbers and Phasors • Step 4: Solve the phasor equation • Step 5: Solve the time domain equation

23. Assignment • Reading:Ulaby, et al. Chapter 2 • Problem Set 1: Some notes • There are 8 problems. Many of the answers to these problems have been provided by either Ulaby or by me. YOU MUST SHOW YOUR WORK TO GET FULL CREDIT! • A key issue in numerical calculation is not presenting more significant figures than you have. You cannot have more significant figures than are in your input data. Please watch that; Ulaby is not careful about it. • Generally, I ask for 3 significant figures, which means that you want to calculate with at least 4. When I want a different number, I tell you. Sometimes, I ask you why I want more.